What is a common-tone diminished seventh chord?

A common-tone diminished seventh chord is a full-diminished seventh chord that has a note in common with its chord of resolution. For example as is common in jazz:

Take the progression in C major:

Em7 - Ebdim7 - Dm7 - G7 - Cmaj7

The Ebdim7 chord has the note A (B double-flat) as a common tone with its chord of resolution (Dm7). If we are talking about seventh chords it actually has two notes in common, A and C.

Generally the diminished seventh chord is found in composition as a leading-tone or secondary leading-tone chord meaning that its function or tendency is to resolve up by half-step to its chord of resolution, just as vii is said to resolve to I in a major key or #vii resolves to i in a minor key.

In the example above, even after trying to re-spell the chord with the four possible "roots", we cannot make the case that this chord has a leading-tone function. This is then described as a common-tone diminished seventh chord for the reason stated earlier.

The real explanation for why this chord works is in the voice-leading.

Note the chromatically descending tenths, E-G (Em7) moving to Eb-Gb (Ebdim7) and finally to D-F (Dm7) as you play these chords on guitar. The chord in reality is simply a way of filling in with four voices a chromatically descending tenth. In jazz you will often hear this in a slightly different way using parallel harmony: Em7 - Ebm7 - Dm7 in which the Ebm7 replaces the Ebdim7 chord. Of course since classical harmony attempts to avoid parallel fifths, the Ebdim7 is a better choice.